Liftings of metallic structures to tangent bundles of order $ r $

Mohammad Nazrul Islam Khan; Uday Chand De; Mohammad Nazrul Islam Khan; Uday Chand De

doi:10.3934/math.2022441

AIMS Mathematics

2022, Volume 7, Issue 5: 7888-7897. doi: 10.3934/math.2022441

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Research article

Liftings of metallic structures to tangent bundles of order $ r $

Mohammad Nazrul Islam Khan ^{1
,
,},
Uday Chand De ²

1.
Department of Computer Engineering, College of Computer, Qassim University, Buraydah, Saudi Arabia
2.
Department of Pure Mathematics, University of Calcutta 35, Ballygaunge Circular Road, Kolkata 700019, West Bengal, India

Received: 31 October 2021 Revised: 31 January 2022 Accepted: 07 February 2022 Published: 17 February 2022
MSC : 53C15, 58A30

It is well known that the prolongation of an almost complex structure from a manifold $ M $ to the tangent bundle of order $ r $ on $ M $ is also an almost complex structure if it is integrable. The general quadratic structure $ F^2 = \alpha F+\beta I $ is a generalization of an almost complex structure where $ \alpha = 0, \; \beta = -1. $ The purpose of this paper is to characterize a metallic structure defined by the general quadratic structure $ F^2 = \alpha F+\beta I, \; \alpha, \beta\in \mathbb{N} $, where $ \mathbb{N} $ is the set of natural numbers. We show that the $ r $-lift of the metallic structure $ F $ in the tangent bundle of order $ r $ is also a metallic structure. Furthermore, we deduce a theorem on the projection tensor in the tangent bundle of order $ r $. Moreover, prolongations of $ G $-structures immersed in the metallic structure to the tangent bundle of order $ r $ and 2 are discussed. Finally, we construct examples of metallic structures that admit an almost para contact structure on the tangent bundle of order 3 and 4.
- lifts,
- metallic structure,
- integrability,
- projection tensor,
- prolongation
Citation: Mohammad Nazrul Islam Khan, Uday Chand De. Liftings of metallic structures to tangent bundles of order $ r $[J]. AIMS Mathematics, 2022, 7(5): 7888-7897. doi: 10.3934/math.2022441

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Abstract

It is well known that the prolongation of an almost complex structure from a manifold $ M $ to the tangent bundle of order $ r $ on $ M $ is also an almost complex structure if it is integrable. The general quadratic structure $ F^2 = \alpha F+\beta I $ is a generalization of an almost complex structure where $ \alpha = 0, \; \beta = -1. $ The purpose of this paper is to characterize a metallic structure defined by the general quadratic structure $ F^2 = \alpha F+\beta I, \; \alpha, \beta\in \mathbb{N} $, where $ \mathbb{N} $ is the set of natural numbers. We show that the $ r $-lift of the metallic structure $ F $ in the tangent bundle of order $ r $ is also a metallic structure. Furthermore, we deduce a theorem on the projection tensor in the tangent bundle of order $ r $. Moreover, prolongations of $ G $-structures immersed in the metallic structure to the tangent bundle of order $ r $ and 2 are discussed. Finally, we construct examples of metallic structures that admit an almost para contact structure on the tangent bundle of order 3 and 4.

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